On the non-existence of tight Gaussian 6-designs on two concentric spheres

A Gaussian t-design is defined as a finite set X in the Euclidean space R-n satisfying the condition: 1/V(R-n) integral(Rn) f(x)e(-alpha 2 parallel to x parallel to 2) dx = Sigma(x is an element of chi) omega(X)f(X) for any polynomial f(x) in n variables of degree at most t, where alpha is a constant real number and omega is a positive weight function on X. It is well known that if X is a Gaussian 2e-design in R-n, then vertical bar X vertical bar >= ((n+e)(e)). We call X a tight Gaussian 2e-design in R-n if vertical bar X vertical bar = ((n+e)(e)). In this paper, we prove that there exists no tight Gaussian 6-design supported by two concentric spheres in R-n for n >= 2. (C) 2013 Elsevier B.V. All rights reserved.